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Full text of "Scientific Papers - Vi"

1913]                       LAMINAR MOTION  OF  AN INVISCID  FLUID                          201
from the second of which we may infer that if q be finite, p+ U must change sign, as we have already seen that it must do when q  0. In every case then, when k is small, the real part of n must equal some value of  kU*.
It may be of interest to show the application of (13) to a case formerly treated j"- in which the velocity-curve is made up of straight portions and is anti-symmetrical with respect to the point lying midway between the two walls, now taken as origin of y. Thus on the positive side
from                 y = 0    to y = $&',          &=>>
from               y=ft' to y = W + b,    Z7 =       + /iF(y -*&');
while on the negative side U takes symmetrically the opposite values.    Then if we write n/kV = ri, (13) becomes
0
+ same with n' reversed. Effecting the integrations, we find after reduction
+ ,u,2626'                          ,,K,
' ............ (    }
in agreement with equation (23) of the paper referred to when k is there made small. Hence n, if imaginary at all, is a pure imaginary, and it is imaginary only when ^ lies between  1/6 and  1/6  2/6'. The regular motion is then exponentially unstable.
In the only unstable cases hitherto investigated the velocity-curve is made up of straight portions meeting at finite angles, and it may perhaps be thought that the instability has its origin in this discontinuity. The method now under discussion disposes of any doubt. For obviously in (13) it can make no important difference whether d U/dy is discontinuous or not. If a motion is definitely unstable in the former case, it cannot become stable merely by easing off the finite angles in the velocity-curve. There exist, therefore, exponentially unstable motions in which both 17 and dU/dy are continuous. And it is further evident that any proposed velocity-curve may be replaced approximately by straight lines as in my former papers.
* By the method of a' former paper " On .the question of the Stability of the Flow of Fluids " (Phil. Mag. Vol. xxxiv. p. 59 (1892) ; Scientific Papers, Vol. m. p. 579) the conclusion that p-i-U must change sign may be extended to the problem of the simple shearing motion between two parallel walls of a viscous fluid, and this whatever may be the value of fc.
t Proc. Land. Math. Soc. Vol. xix. p. 67 (1887); Scientific Papers, Vol. in. p. 20, figs. (3), (4), (5).e velocities vanish at the boundary. The motion M' with velocities u', v', w' does not in general satisfy the dynamical equations. We have